Worthington is great for families!
Although it's perceived as a place with above-average incomes, you'll hardly
ever hear Worthington folks talk about that. It's definitely not like Upper Arlington or Dublin—preoccupied with wealth and status. Worthingtonians are generally much more laid back.
Don't forget, there are areas outside of Worthington proper that still belong to the Worthington school district (zip code 43235). You may find more affordable homes in any of those neighborhoods as well. Newer housing stock there, too—mostly post-1980s.
Of course, you've probably already noticed the quaintness of Worthington's historic downtown. They really take pride in their heritage, and keeping the town beautiful.
Clintonville is sometimes called the "suburb that's in the city" since it is actually part of incorporated Columbus. Your kids would go to Columbus schools there (

) unless you plan on sending them to private schools. Also, Clintonville is
a lot more liberal than Worthington. Many Clintonville people hold beliefs and practice lifestyles that are at variance with traditional American values. If that's OK with you, then living there shouldn't be a problem. But it's worth being forewarned...
If I were raising children, I'd much prefer Worthington over the snobbier suburbs or the liberal quasi-urban neighborhoods.